Most of the authors of research papers in compressed sensing use n-widths and Kolmogorov's entropy extensively, which are kind of hard for me to understand. Any suggestion on books or expository articles about these will be highly appreciated.

Nothing can surpass the Li and Vitányi book in both readability and comprehensiveness,
but that is a significant undertaking, and you might need alternatives.
"A Short Introduction to Kolmogorov Complexity" by Volker Nannen, true to its title, is only 7 pages long. The Wikipedia page on Kolmogorov Complexity is quite good.
Gregory Chaitin's Exploring Randomness is a fun, quirky, personalized view of the field, emphasizing his own work and LISP programs. Caveat: his philosophical musings are quite controversial.
In fact, reading the balanced Notices of the AMS review of his book by Panu Raatikainen might serve as a useful introduction to the area.